Application of collocation and orthogonality-based techniques to satisfy junction conditions in an arbitrary...

The pressure field within adjacent sections of a waveguide are related by junction conditions, which require that the pressure and particle velocity normal to the shared part of the junction be continuous. The formulation is general, limited only by the requirement that prior analysis of each segment has identified a set of transverse mode functions. The derivation begins with consideration of a generic boundary condition. It is proven that a collocation procedure is the same as a numerical approximation of the integrals in an analysis that uses modal orthogonality to minimize the junction condition errors. The collocation concept is used to derive algebraic equations relating the propagation functions for the connected waveguide segments. A wide range of configurations that can be analyzed with the collocation formulation are shown to not fit the requirements for an orthogonality-based analysis. An observation that Euler's equation requires that the particle velocity tangent to the junction be continuous leads to a recommendation that this property should be used to verify any analysis.